Binomial Coefficient with Self

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Theorem

$\forall n \in \Z: \dbinom n n = \left[{n \ge 0}\right]$

where $\left[{n \ge 0}\right]$ denotes Iverson's convention.

That is:

$\forall n \in \Z_{\ge 0}: \dbinom n n = 1$
$\forall n \in \Z_{< 0}: \dbinom n n = 0$


Proof

From the definition of binomial coefficient:

$\dbinom n n = \dfrac {n!} {n! \ \left({n - n}\right)!} = \dfrac {n!} {n! \ 0!}$

the result following directly from the definition of the factorial, where $0! = 1$.

From N Choose Negative Number is Zero:

$\forall k \in \Z_{<0}: \dbinom n k = 0$

So for $n < 0$:

$\dbinom n n = 0$

$\blacksquare$


Also see


Sources